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LECTURE 1. INVARIANTS AND DISCRIMINANTS OF PLANE CURVES Preface to Lecture 1 CHAPTER 1. Plane Curves 1. The three basic invariants 2. Properties of the basic invariants 3. Computation of basic invariants 4. Extremal curves and tree-like curves 5. The numerology 6. Cobordisms 7. Long curves CHAPTER 2. Legendrian Knots 8. From plane curves to Legendrian knots 9. The space of Legendrian curves 10. The basic invariant J+ 11. The Legendrian linking numbers 12. Calculation of linking numbers LECTURE 2. SYMPLECTIC AND CONTACT TOPOLOGY OF CAUSTICS AND WAVE FRONTS, AND STURM THEORY CHAPTER 3. Singularities of Caustics and Sturm Theory 13. The Lagrangian collapse and the last geometrical theorem of Jacobi 14. The four cusp theorem 15. Sturm theory and Morse theory 16. Proof of the four cusps theorem CHAPTER 4. Singularities of Wave Fronts and the Tennis Ball Theorem 17. The wave fronts reversal 18. The front reversal four cusps theorem 19. The proof of the existence of four cusps on a front 20. The tennis ball theorem References